The present invention relates to radar systems and, in particular, to systems adapted for `clutter-type` targets such as clouds.
A striking feature of the radar return from clouds and other "clutter" type targets is the large pulse-to-pulse fluctuation in the received power. One obvious reason for the large fluctuations is that the returns are derived from cloud particles which are randomly-arranged and constantly changing. The same is true for other clutter type targets such as the detection of an aircraft hidden in chaff or small targets obscured by clutter echoes from the sea. The present radar system contemplates all such targets but, since it is primarily concerned with clouds or related meteorological applications, the following description is directed toward these particular applications.
A clearer understanding of the present problem can be obtained by considering a conventional meteorological radar which projects a pulse of a particular, selected frequency leaving the transmitter at time (T) and being backscattered by the ith cloud particle at range (R.sub.1). The scattering amplitude at the receiver can be written: EQU a.sub.i = a.sub.oi sin[ (.omega.o + .omega.i) t+ .phi..sub.oi ],
Where a.sub.oi is the square root of the particle scattering cross-section, and .omega.o is the angular frequency of the transmitter of wavelength .lambda.. The term .omega..sub.i = 4.pi.v.sub.i /.lambda. is the Doppler angular frequency associated with the ith particle whose component of velocity along the radar beam is v.sub.i. The phase angle .phi..sub.oi referred to the transmitter is 4.pi.r.sub.oi /.lambda..
For an extended target such as a cloud, the signal reaching the receiver at a particular instant is the sum of the backscattered signals from all the particles contained in a volume whose length is one-half the pulse length and of cross-section equal to the area illuminated by the beam.
With a receiver using conventional square law envelope detection, the instantaneous power is proportional to the square of the amplitude, and is given by: ##EQU1## where A is the sum of the individual scattering amplitudes.
The first sum over all the scattering cross-sections is proportional to the average received power, or the mean intensity, and is designated by A.sup.2. It is this quantity which is desired as a measure of cloud reflectivity or rain intensity.
The second term containing the cosine represents the fluctuations or deviations from the mean, and is due to the relative velocities of the scatterers. This "phase noise" term can be interpreted as the spectrum of Doppler beat frequencies resulting from the finite width of the particle velocity spectrum. Note that if .omega..sub.i =.omega..sub.j, i.e., if the particle velocity components v.sub.i and v.sub.j along the beam are equal, the fluctuation rate goes to zero. Thus, the broader the size distribution, the greater will be the fluctuation rate.
Since a single radar pulse samples only one particular random arrangement of cloud particles at any given instant, the expectation value for the cosine term is not zero. A second pulse at a later time will sample another of an infinite number of possible particle arrangements, thus providing a second value for the total backscattered power. It is thus not possible to specify the backscattered intensity precisely. The probability distribution of the intensity A.sup.2 for a single measurement is given by EQU P(A.sup.2)dA.sup.2 = (1/ A.sup.2)e.sup.-.sup.A.spsp.2/A.spsp.2 dA.sup.2
where A.sup.2 is the mean or true value of intensity. The important consideration to be derived from this relationship is that the root mean square (RMS) error in the estimate of the intensity from a single pulse is equal to the mean, A.sup.2. In other words, the RMS error is so large that the estimate of A.sup.2 from a single pulse can be either double the mean or zero since, as noted, the error is equal to the mean, A.sup.2.
The above discussion shows that a single measurement is essentially useless as an estimate of cloud reflectivity or rain intensity. In meteorological radars it is therefore customary to average the returns from a large number of independent samples. It is precisely at this point that conventional weather radars, which transmit a single frequency pulse, suffer a constraint which severely limits their scanning speed.
Generally, the number of independent samples which are averaged to improve the estimate of mean reflected power is chosen as a compromise between the need for following rapidly changing phenomena, such as vigorous thunderstorms, and the desired accuracy of the reflectivity estimate. The standard deviation (rms error) for an average of k independent samples of intensity is: A.sup.2 /.sqroot.k. To obtain an intensity estimate with a standard deviation of 10 percent, it is necessary to average 100 independent samples. Averaging 15 independent samples gives a standard deviation of 25 percent.
The rate at which independent samples may be obtained from clouds is not simply a function of the pulse repetition frequency. If it were, increasing the PRF of the radar consistent with the desired unambiguous range would in some cases remove the scanning speed limitation. Unfortunately, the limitation is imposed primarily by the distribution in fall velocity of the scatterers which determines the rate of rearrangement, and hence, the time interval necessary before a new pulse will provide an independent sample. This can be seen by again examining the cosine term in equation (1); the fluctuation rate is determined by the distribution of particle velocities, or the width of the Doppler spectrum. As an example, for snow, whose Doppler spectrum exhibits a width of about 0.3 M/sec, an independent sample is possible every 15 msec with a 3 cm radar. For a 10 cm radar, the time to independence is approximately 50 msec. On the other hand, in a vigorous thunderstorm, the width of the Doppler spectrum may be as high as 8 or 10 M/sec, so that a 3 cm radar will obtain an independent sample every millisecond.
The nature of the problem also becomes very clear when calculations of the required scanning rate of such single frequency radars is considered. Thus, calculations can be made of the maximum scanning rate for a single frequency radar of 3 cm wavelength (9000 MHz) and an antenna bandwidth of 1.5.degree.. Choosing the most favorable case of one millisecond for the time to independence (PRF = 1 KHz), and asking that the reflectivity estimate lie within .+-. 20 percent of the true mean (.+-. 1db), it can be calculated that averaging 25 samples per beamwidth will be necessary. It will take 360 sec, or 6 minutes, to scan a hemisphere of sky. In view of the average thunderstorm cell life of about 25 minutes, four samples per storm is hardly adequate to study the dynamics of growth. Even one sample of sky which takes 6 minutes to obtain can hardly be called a "snapshot". The situation is obviously much worse for snow. For a 10 cm radar of the same beamwidth, it will take about 20 minutes to scan a hemisphere.